Coxeter Groups Unify Polytopes, Tessellations, and Molecular Symmetries — E8 Intelligence Research

FINDING: Coxeter groups provide the complete algebraic classification of all regular polytopes and reflection symmetries in any dimension, unifying polyhedra, tessellations, and molecular symmetries under a single combinatorial framework. | MATH: Coxeter group presentation: ⟨r₁,…,rₙ | (rᵢrⱼ)^(mᵢⱼ)=1⟩ with mᵢᵢ=1, mᵢⱼ≥2; finite classification yields ADE (simply-laced), BCF (non-simply-laced), H₃, H₄, I₂(p); Dynkin diagrams encode mᵢⱼ=3,4,6 for edges; Cartan matrix entries: 2cos(π/mᵢⱼ); root system lengths ratio √2 for Bₙ, φ for H₃/H₄; Coxeter number h = 2|R|/n. | CONNECTION: H₃ (icosahedral) and H₄ (600-cell/120-cell) groups involve golden ratio φ = (1+√5)/2 ≈ 1.618, with dihedral angles 2π/5 (72°) and cos(π/5)=φ/2; ratios 0.618 (1/φ), 0.382 (1/φ²), 0.786 (√(φ/2)?) appear in H₃ root system; E₈ lattice (Coxeter number 30) relates to quasicrystalline symmetries; base-60 appears in Babylonian astronomy but not directly in Coxeter classification; crystallographic Coxeter groups (Weyl groups) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com